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The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 5, October 2013
- pp. 1271-1290
- 10.1353/ajm.2013.0040
- Article
- Additional Information
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Let $(M,g)$ be a three-dimensional smooth compact Riemannian manifold such that
all geodesics are simple and closed with a common minimal period, such as the
$3$-sphere ${\Bbb S}^3$ with canonical metric. In this work the global well-posedness
problem for the quintic nonlinear Schr\"odinger equation $i\partial_t u+\Delta u=
\pm |u|^4u$, $u|_{t=0}=u_0$ is solved for small initial data $u_0$ in the energy
space $H^1(M)$, which is the scaling-critical space. Further, local well-posedness
for large data, as well as persistence of higher initial Sobolev regularity is
obtained. This extends previous results of Burq-G\'erard-Tzvetkov to the endpoint
case.